Rotation Group SO Research Articles

A unified approach to scattered data approximation on $\mathbb{S}^{\bf 3}$ and SO(3)

In this paper we use the connection between the rotation group SO(3) and the three-dimensional Euclidean sphere $\mathbb{S}^{3}$ in order to carry over results on the sphere $\mathbb{S}^{3}$ directly to the rotation group SO(3) and vice versa. More precisely, these results connect properties of sampling sets and quadrature formulae on SO(3) and $\mathbb{S}^{3}$ respectively. Furthermore we relate Marcinkiewicz---Zygmund inequalities and conditions for the existence of positive quadrature formulae on the rotation group SO(3) to those on the sphere $\mathbb{S}^{3}$ , respectively.

A minimal atlas for the rotation group SO(3)

We describe explicitly an atlas for the rotation group SO(3) consisting of four charts where each chart is defined by Euler angles or each chart is defined by Cardan angles. This is best possible since it is well known that three charts do not suffice.

Inversion of the noisy Radon transform on [formula omitted] by Gabor frames and sparse recovery principles

The inversion of the one-dimensional Radon transform on the rotation group SO ( 3 ) is an ill-posed inverse problem that can be applied to X-ray tomography with polycrystalline materials. This paper is concerned with the development of a method to stably approximate the inverse of the noisy Radon transform on SO ( 3 ) . The proposed approach is composed by basic building blocks of the coorbit theory on homogeneous spaces, Gabor frame constructions and variational principles for sparse recovery. The performance of the finally obtained iterative approximation is studied through several experiments.

On Coordinate Independent State Space of Matrix Theory

We investigate symmetry structure of coordinate independent states in Matrix theory. Those states are building blocks of gauge invariant wavefunctions, and especially we consider zero energy bound state wavefunctions. First we construct some states in lower representations of the space rotation group SO(9) explicitly in the case of SU(2) gauge group, and classify them into SU(2) representations. Next we count the multiplicities of SO(9)×SU(N) representations in the coordinate independent state space by using the notion of characters in group theory. For N = 2 case we give complete decomposition of the space into SO(9)×SU(2) representations, and give partial results for higher gauge groups.

Probabilistic Marcinkiewicz–Zygmund Inequalities on the Rotation Group

Approximation problems on the rotation group SO(3) naturally arise in various fields, like crystallography, chemistry, and biology. For example, in crystallographic texture analysis one is confronted with the problem of evaluating so-called orientation density functions (ODFs). In many situations one only has a finite number of measurements at scattered sampling nodes. In order to reconstruct ODFs over all rotations, so-called Marcinkiewicz–Zygmund inequalities on the rotation group are an important tool. These inequalities provide norm equivalences between polynomials on SO(3) and their sample values. Recently shown equivalences depend on a density parameter of the sampling set and the proven inequalities hold true for polynomials on SO(3) whose degree does not exceed an upper bound which is determined by this density parameter. In this paper, we show that we can enlarge this upper bound for the polynomial degree significantly if we are satisfied by such norm equivalences that hold with a given probability only. Moreover, we show that there are fixed sampling sets for which we get probabilistic Marcinkiewicz–Zygmund inequalities that hold for polynomials on SO(3) of all degrees.

Wavelets onS3andSO(3)-Their construction, relation to each other and Radon transform of wavelets onSO(3)

The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S3 and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L2-functions on S3. Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S2 for every fixed detection direction. Copyright © 2010 John Wiley & Sons, Ltd.

A MATHEMATICAL REVISION OF THE LANDAU-LIFSHITZ EQUATION

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M(superscript 3+1) has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SO0(3,1), and the numerical method based on the internal symmetry was developed in [29]. This paper derives another four new representations of the Landau-Lifshitz equation. We prove that this equation admits two generators: one conservative and one dissipative, as well as two brackets: Poisson bracket and dissipative bracket. Upon embedding the Landau-Lifshitz equation into a skew-symmetric matrix space, we can develop a double-bracket flow representation. The conserved magnetization' magnitude is just the result of the isospectrality for an iso-spectral flow equation. Finally, on the cotangent bundle of an invariant manifold of the constant magnetization magnitude, we introduce the Lie-Poisson bracket to construct an evolutional differential equations system. The magnetization trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the rotation group SO(3). The six different representations including the one by Bloch et al. [3] are compared.

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature 𝕊2 × 𝕊1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on 𝕊2 and 𝕊1, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on 𝕊2 and 𝕊1 lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.

$G_2$ and the rolling distribution

Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphere. But if the ratio of radii of the spheres is 1:3 and if the distribution is lifted to the universal cover S^3 \times S^2 of M, then the symmetry group becomes much larger: the split real form of the Lie group G_2. This fact goes back to Cartan in a sense, and can be found in a paper by Bryant and Hsu. We prove this fact through two explicit constructions, relying on the theory of roots and weights for the Lie algebra of G_2, and on its 7-dimensional representation.

Nonlinear subelliptic equations

In this paper, we study the higher order regularity for weak solutions of a class of quasilinear subelliptic equations. We introduce the notion of ν-closed Hormander system of vector fields, which includes all the previously studied nilpotent systems and extends them to some classes of non-nilpotent systems of vector fields, including those generating the Lie Algebra of the rotation group SO(n) and other non-compact semisimple and solvable Lie groups.

Ultimate braid-group generators for coordinate exchanges of Ising anyons from the multi-anyon Pfaffian wavefunctions

We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd numbers of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu–Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions are completely equivalent to the spinor representations of SO(2n + 2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2n chiral Ising spin fields σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.

Tensor coordinates in noncommutative mechanics

A consistent classical mechanics formulation is presented in such a way that, under quantization, it gives a noncommutative quantum theory with interesting new features. The Dirac formalism for constrained Hamiltonian systems is strongly used, and the object of noncommutativity θij plays a fundamental rule as an independent quantity. The presented classical theory, as its quantum counterpart, is naturally invariant under the rotation group SO(D).

A fast algorithm for nonequispaced Fourier transforms on the rotation group

In this paper we present algorithms to calculate the fast Fourier synthesis and its adjoint on the rotation group SO(3) for arbitrary sampling sets. They are based on the fast Fourier transform for nonequispaced nodes on the three-dimensional torus. Our algorithms evaluate the SO(3) Fourier synthesis and its adjoint, respectively, of B-bandlimited functions at M arbitrary input nodes in \(\mathcal O(M+B^4)\) or even \(\mathcal O(M + B^3 \log^2 B)\) flops instead of \(\mathcal O(MB^3)\). Numerical results will be presented establishing the algorithm’s numerical stability and time requirements.

The structure of supercooled liquids and glasses as Skyrmionic texture

A re-appraisal of the frustration models for glass-forming and supercooled liquids is presented. It is shown that the elasticity theory expressed by covariant derivatives with respect to an ideal template on a three-sphere S3 allows for the formation of localized soft Skyrmion states. Owing to the homomorphism between the proper rotation group SO(4) for the symmetry of S3 and the SU(2) group for the unitary 2×2 matrices, the simplest version of the frustration models contain Skyrme’s hedgehog solution stabilized by the fixed gauge background from the covariant derivatives. The structure of glasses and supercooled liquids can be rendered as dense Skyrmionic matter which reproduces geometric frustration at larger length scales in an orientable continuum.

Error estimates for kernel and projection methods of recovering the orientation distribution function on SO(3)

The orientation density function is recovered from a sample of orientations on the rotation group SO(3) of the three-dimensional Euclidean space. Sufficient conditions for the consistency of kernel and projection estimates in L2, L1, and C are considered. Numerical results concerning the error estimation of projection methods over the basis of generalized spherical functions are given for normal distributions on SO(3).

Subrepresentation semirings and an analog of 6j-symbols

Let V be a complex representation of the compact group G. The subrepresentation semiring associated to V is the set of subrepresentations of the algebra of linear endomorphisms of V with operations induced by the matrix operations. The study of these semirings has been motivated by recent advances in materials science, in which the search for microstructure-independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group SO(3). In this case, the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j-symbols familiar from the quantum theory of angular momentum. In this paper, we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by Mackey [“Multiplicity free representations of finite groups,” Pac. J. Math. 8, 503 (1958)]. We introduce a new class of symbols called twisted 6j-symbols for these groups, and we explicitly calculate the structure constants for subrepresentation semirings in terms of these symbols. Moreover, we show that these symbols satisfy analog of the standard properties of classical 6j-symbols.

Stability results for approximation by positive definite functions on SO(3)

We consider interpolation methods defined by positive definite functions on a locally compact group G. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented, and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally, we apply our results to construct well-localized positive definite basis functions having nice stability properties on the rotation group SO ( 3 ) .

A quotient group of the group of self homotopy equivalences of SO(4)

The author studies the quotient group $\mathscr{E}$(SO(4))/$\mathscr{E}$#(SO(4)), where $\mathscr{E}$(SO(4)) is the group of homotopy classes of self homotopy equivalences of the rotation group SO(4) and $\mathscr{E}$#(SO(4)) is the subgroup of it consisting of elements that induce the identity on homotopy groups.

FFTs on the Rotation Group

We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B4) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B3)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B6) algorithm derived from a basic quadrature rule on O(B3) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B3log 2B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well.

The Radon transform on SO(3): a Fourier slice theorem and numerical inversion**Dedicated to Professor Manfred Tasche on the occasion of his 65th birthday.

The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Radon transform of a function sampled on the product space of two two-dimensional spheres is determined as the solution of a minimization problem, which is iteratively solved using fast Fourier techniques for and SO(3). The favorable complexity and stability of the algorithm based on these techniques has been confirmed with numerical tests.